Attaching Hurewicz fibrations with fiber preserving maps

Arnold, James E.

doi:10.2140/pjm.1973.46.325

Cited by 4 publications

(6 citation statements)

References 8 publications

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“…This follows from the references and arguments above. We remark that f : B 1 → B 2 being a fibration implies that Y 1 → B 1 → B 2 is a fibration so that [1] applies to show that Y 2 → B 2 is a fibration.…”

Section: A Convenient Model For A-theorymentioning

confidence: 99%

The cobordism category and Waldhausen’s 𝐾-theory

Bökstedt¹,

Madsen²

2014

Contemporary Mathematics

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Section: A Convenient Model For A-theorymentioning

confidence: 99%

The cobordism category and Waldhausen’s 𝐾-theory

Bökstedt¹,

Madsen²

2014

Contemporary Mathematics

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“…(1) p is a bundle of n-dimensional compact (not necessarily closed) manifolds (that is, p is the projection map of a fibre bundle with fibres n-dimensional compact topological manifolds), (2) ϕ is a homotopy equivalence, and (3) E is a subset of B × U such that p corresponds to the projection from B × U to B.…”

Section: Structure Spaces On Fibrationsmentioning

confidence: 99%

Higher Whitehead torsion and the geometric assembly map

Steimle

2012

Journal of Topology

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We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a 'geometric assembly map' defined on stabilized structure spaces in purely homotopy-theoretic terms. IntroductionGiven a space X, the structure space S n (X) is the space of all homotopy equivalences M → X, where M is an n-dimensional manifold (of some given type, such as compact or closed, differentiable, piecewise-linear, or topological). Obviously, X has the homotopy type of such a manifold if and only if the structure space of X is non-empty. In this case, π 0 S n (X) is the central object of interest for the classification of manifolds in the homotopy type of X; the higher homotopy type of S n (X) is closely related to the study of automorphisms of these and to the classification of families of manifolds homotopy equivalent to X [23].If p : E → B is a given fibration, then the above construction can be generalized as follows: A point in the structure space S n (p) of p is given by a bundle of n-dimensional manifolds E → B over B together with a fibre homotopy equivalence E → E. (So the structure space of the space X is the structure space of the trivial fibration X → * .) Pull-back defines a pairing S n (B) × S k (p) −→ S n+k (E), thus if B comes with a given structure, then evaluation of this pairing defines a map α : S k (p) −→ S n+k (E).Geometrically, this map assembles all the manifold structures on the individual fibres to one manifold structure on E, so we call it the geometric assembly map. It is relevant, for instance, in the study of fibring questions.Algebraic K-theory of spaces [19] is a key tool in understanding families of manifolds. The connection to the topology of manifolds is established by the stable parametrized h-cobordism theorem [20], which classifies families of h-cobordisms in terms of K-theory, in a stable range. Recently, Hoehn [7] has used the stable parametrized h-cobordism theorem to describe the homotopy type of the stabilized structure space S ∞ (p) := colim S n (p)

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“…Lemma A.6 implies Y | B0 → Y is a cofibration and hence Y 0 → Y | B0 is also a cofibration [Str72, Lemma 5]-the lemma in[Str72] is stated in the category T, yet in view of Lemma A.6, the same assertion holds in the category Top w . Secondly, by Lemma 3.8 in[JEA73] , we know there is a retractionr : E| B0 × I → Y | B0 ∪ Y 0 × I (35) over B 0 [HK78, Definition 1.1]. Again, in[HK78] and[JEA73], they are working in the category T. However, their approaches can be applied also to the category Top w .…”

mentioning

confidence: 99%

“…Secondly, by Lemma 3.8 in[JEA73] , we know there is a retractionr : E| B0 × I → Y | B0 ∪ Y 0 × I (35) over B 0 [HK78, Definition 1.1]. Again, in[HK78] and[JEA73], they are working in the category T. However, their approaches can be applied also to the category Top w . In more details, one observes that, similar to the case in T [JEA73, p.325],a map E q − → L in Top w is a Hurewicz fibration in Top w if and only if there exists a universal section of the mapE I γ →(γ(0),q•γ) − −−−−−−−− → E × L L I ,where E × L L I is given by the limitlim(E q − → L ev0 ← − − L I )and ev 0 (β) = β(0).…”

mentioning

confidence: 99%

“…In more details, one observes that, similar to the case in T [JEA73, p.325],a map E q − → L in Top w is a Hurewicz fibration in Top w if and only if there exists a universal section of the mapE I γ →(γ(0),q•γ) − −−−−−−−− → E × L L I ,where E × L L I is given by the limitlim(E q − → L ev0 ← − − L I )and ev 0 (β) = β(0). This observation implies that one can attach Hurewicz fibrations in Top w as in T [JEA73, Lemma 3.8]-the argument in[JEA73, Lemma] works also in the category Top w . Therefore, the map Y | B0 ∪ Y 0 × I → B 0 is also a Hurewicz fibration in Top w , and the diagram belowM (C) × O(C) M (C) × O(C) M (C) M (C) × O(C) M (C) M (C) × O(C) M (C) ) × O(C) M (C) M (C) × O(C) O(C) M (C) × O(C) M (C) M (C) ec × O(C) id id × O(C) ec •c p1 p2where p 1 and p 2 are obvious projections to the first component and the second component, respectively.…”

mentioning

confidence: 99%

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Categories of vector spaces and Grassmannians

Wang

2017

Preprint

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Attaching Hurewicz fibrations with fiber preserving maps

Cited by 4 publications

References 8 publications

The cobordism category and Waldhausen’s 𝐾-theory

The cobordism category and Waldhausen’s 𝐾-theory

Higher Whitehead torsion and the geometric assembly map

Categories of vector spaces and Grassmannians

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